A set of relative prime, positive integers $A = [a_1, \dots, a_d]$ describe the restricted partition function
$$
p_A(n) = \# \{(m_1,\dots,m_d)\in\mathbb{Z}^d: \textrm{ all }m_j \geq 0, \sum_{j=1}^d m_j a_j = n \}
$$
I'm interested in a binary version of $p_A(n)$, namely
$$
q_A(n) =
\begin{cases}
1 \textrm{ if } p_A(n) \geq 1 \\
0 \textrm{ if } p_A(n) = 0
\end{cases}
$$
Let $P_A(n)$ and $Q_A(n)$ be the polynomial parts of $p_A(n)$ and $q_A(n)$, respectively. In fact, $p_A(n)$ is polynomial except the last coefficient term. Let us interpret $P_A(n)$ as the expected number of solution to the restricted partition and $Q_A(n)$ as the probability that there is a solution. *Without* the underlying structure, we could relate these quantities by:
$$
Q_A(n) = 1 - \exp ( - P_A(n) )
$$

Given a specified $\alpha$, how to choose $A$ with $\alpha \approx \prod_{j=1}^N a_j$ such that $Q_A(n)$ is close to $1 - \exp ( - P_A(n) )$?

Example: $A = [49,51,100]$ is a bad choice, because $100$ gets "eaten" up by $49$ and $51$. Whereas $A = [49,63,81]$ is a better choice.

As is apparent, I am missing quite some terminology here, for example is there a name for $q_A(n)$?